Answer:
[tex]f'(x)=\frac{1}{xln(x)}[/tex]
Step-by-step explanation:
In order to solve this problem, we rewrite the original function as:
[tex]f(x)=ln(3 ln(x))[/tex]
In order to find the derivative, we apply the chain rule for a composite function, which states that:
[tex]\frac{d}{dx}g(f(x))=\frac{dg}{df}\cdot \frac{df}{dx}[/tex]
By applying the chain rule, we obtain the following:
[tex]\frac{d}{dx}f(x)=\frac{1}{3ln(x)} \cdot \frac{d}{dx}(3ln(x))[/tex] (1)
We also know that the derivative of the logarithm is
[tex]\frac{d}{dx}ln(x)=\frac{1}{x}[/tex]
And since 3 is just a constant, expression (1) becomes:
[tex]\frac{d}{dx}f(x)=\frac{1}{3ln(x)} \cdot \frac{d}{dx}(3ln(x))=\frac{1}{3ln(x)}\cdot \frac{3}{x}=\frac{1}{xln(x)}[/tex]
So, the derivative of the function is
[tex]f'(x)=\frac{1}{xln(x)}[/tex]
